let M be non empty set ; :: thesis: for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V holds f1 + f2 = f2 + f1
let V be ComplexNormSpace; :: thesis: for f1, f2 being PartFunc of M,the carrier of V holds f1 + f2 = f2 + f1
let f1, f2 be PartFunc of M,the carrier of V; :: thesis: f1 + f2 = f2 + f1
A1: dom (f1 + f2) = (dom f2) /\ (dom f1) by Def1
.= dom (f2 + f1) by Def1 ;

now

let x be Element of M; :: thesis: ( x in dom (f1 + f2) implies (f1 + f2) /. x = (f2 + f1) /. x )
assume A2: x in dom (f1 + f2) ; :: thesis: (f1 + f2) /. x = (f2 + f1) /. x
hence (f1 + f2) /. x = (f2 /. x) + (f1 /. x) by Def1
.= (f2 + f1) /. x by A1, A2, Def1 ;
:: thesis: verum

end;

hence f1 + f2 = f2 + f1 by A1, PARTFUN2:3; :: thesis: verum